A Polarization Problem

I'm trying to teach myself polarization and dielectrics by doing problems, but it's not going as well as I'd hoped. Here's the first problem that I got stuck on:

A round dielectric disc of radius R is statically polarized so that it gains the uniform polarization P, with the vector P lying in the plane of the disc. Find the strength E of the electric field at the center of the disc if d << R. ​

I thought that since the disc is polarized along its plane, there would be a charge density [tex]\sigma[/tex] on one half of it, and a charge density of [tex]-\sigma[/tex] on the other half of it. I wasn't sure if [tex]\sigma = P[/tex], but I set them equal anyway. I also wasn't sure if [tex]\sigma[/tex] is uniform along each half-surface, but I did that anyway. Then I thought that a uniform polarization vector means that the whole surface is charged up unformly, which means no net electric field at the center.

I realize this is going to sound silly, but I have no idea how I'm supposed to solve this problem. Where am I supposed to start?